for any P and R, because we can take φ(x,y) to be x=y. Now,
suppose (1) holds. Take φ(x,y) to be

(5) P is to x as R is to y.

Then (4) and (1) together yield (2). For example, Davis Square
is to Flushing as Boston is to New York, because Davis Square is to
Davis Square as Flushing is to Flushing, and
Davis Square is to Boston as Flushing is to New
York. QED.

This proof is problematic: What is a “sentence with two holes”,
and how can we splice things such as fish and bicycles into
sentences, so as to draw analogies involving fish and bicycles (as
opposed to the words “fish” and “bicycles”, which neither swim nor
roll)? For that matter, what (sortof
parametricity) stops us from taking φ(x,y) to be